Can you Move and then make a second Move which ends in a Jump which ends in getting into base to base contact and thus close combat? Or does it turn into a Run and thus prohibit you from entering into base to base? I'm almost sure that it is the latter but wanted to make sure.

But the more important question: can you Jump without a need to jump? So instead of simply Moving into combat could you move part of the way and then Jump the last two Inches gaining the Charge and the Jump bonuses? This seems like it would be pretty trivial for a high Finesse Character since the difficulty of the 2" Jump is so low? And it would allow you to use your first Action for something different and still get the Charge bonus (plus the additional Jump bonus).

Question 1: If the move could be considered to be in a straight line I'd allow a charge. So, if you are on a roof and can make a straight line to your foe then I'd allow a charge which includes the jump - a kind of wild, diving charge admittedly but a charge nonetheless. This wouldn't give you any extra bonus though as the jump, in itself, is considered a kind of charge. If your move isn't a straight line then no - you can't engage. So in essence, you are correct
Having said that if you only need a single action to get to the point of the jump then you could engage. I know the rules kind of suggest otherwise but I figure in the spirit of fun (and spectacular actions) I'd allow it. I will update the FAQ for this case I think

Question 2: No. You must jump from an elevated position. Just springing into contact from ground level, whilst acrobatic, wouldn't give you the bonuses which are designed to represent the shock and relative force of someone leaping onto you from above.